370:
metal, and the interest lies in analyzing this spatial change of temperature within the object over time until all gradients disappear entirely (the ball has reached the same temperature as the oil). Mathematically, this condition is also approached exponentially; in theory, it takes infinite time, but in practice, it is over, for all intents and purposes, in a much shorter period. At the end of this process with no heat sink but the internal parts of the ball (which are finite), there is no steady-state heat conduction to reach. Such a state never occurs in this situation, but rather the end of the process is when there is no heat conduction at all.
4921:, such as evaporation or fusion, at the temperature at which it must conduct heat. But when only thermal equilibrium is considered and time is not urgent, so that the conductivity of the material does not matter too much, one suitable heat conductor is as good as another. Conversely, another aspect of the zeroth law is that, subject again to suitable restrictions, a given diathermal wall is indifferent to the nature of the heat bath to which it is connected. For example, the glass bulb of a thermometer acts as a diathermal wall whether exposed to a gas or a liquid, provided that they do not corrode or melt it.
3938:
1280:
1173:
1064:
5414:
366:
steady-state, in which a constant temperature gradient along the bar is finally set up, and this gradient then stays constant in time. Typically, such a new steady-state gradient is approached exponentially with time after a new temperature-or-heat source or sink, has been introduced. When a "transient conduction" phase is over, heat flow may continue at high power, so long as temperatures do not change.
4888:, varies in practice. In steel, the quenching temperature range is generally from 600 °C to 200 °C. To control the quenching time and to select suitable quenching media, it is necessary to determine the Fourier number from the desired quenching time, the relative temperature drop, and the relevant Biot number. Usually, the correct figures are read from a standard
1019:
4102:. If the system has a Biot number of less than 0.1, the material behaves according to Newtonian cooling, i.e. with negligible temperature gradient within the body. If the Biot number is greater than 0.1, the system behaves as a series solution. The temperature profile in terms of time can be derived from the equation
2283:
362:. In this state of steady-state equilibrium, temperatures vary greatly from the engine cylinders to other parts of the automobile, but at no point in space within the automobile does temperature increase or decrease. After establishing this state, the transient conduction phase of heat transfer is over.
4965:
Working: if the same gas is present around all the
Wheatstone bridge filaments, then the same temperature is maintained in all the filaments and hence same resistances are also maintained; resulting in a balanced Wheatstone bridge. However, If the dissimilar gas sample (or gas mixture) is passed over
345:
Another term is "non-steady-state" conduction, referring to the time-dependence of temperature fields in an object. Non-steady-state situations appear after an imposed change in temperature at a boundary of an object. They may also occur with temperature changes inside an object, as a result of a new
4953:
containing four filaments whose resistances are matched. Whenever a certain gas is passed over such network of filaments, their resistance changes due to the altered thermal conductivity of the filaments and thereby changing the net voltage output from the
Wheatstone Bridge. This voltage output will
411:
during transient cooling (or the reverse during heating). The equivalent thermal circuit consists of a simple capacitor in series with a resistor. In such cases, the remainder of the system with a high thermal resistance (comparatively low conductivity) plays the role of the resistor in the circuit.
4969:
Using this technique many unknown gas samples can be identified by comparing their thermal conductivity with other reference gas of known thermal conductivity. The most commonly used reference gas is nitrogen; as the thermal conductivity of most common gases (except hydrogen and helium) are similar
484:
in the temperature and to the area, at right angles to that gradient, through which the heat flows. We can state this law in two equivalent forms: the integral form, in which we look at the amount of energy flowing into or out of a body as a whole, and the differential form, in which we look at the
357:
An example of a new source of heat "turning on" within an object, causing transient conduction, is an engine starting in an automobile. In this case, the transient thermal conduction phase for the entire machine is over, and the steady-state phase appears, as soon as the engine reaches steady-state
206:
In gases, heat transfer occurs through collisions of gas molecules with one another. In the absence of convection, which relates to a moving fluid or gas phase, thermal conduction through a gas phase is highly dependent on the composition and pressure of this phase, and in particular, the mean free
124:
Conduction is the main mode of heat transfer for solid materials because the strong inter-molecular forces allow the vibrations of particles to be easily transmitted, in comparison to liquids and gases. Liquids have weaker inter-molecular forces and more space between the particles, which makes the
424:
is a model that is compatible with the theory of special relativity. For most of the last century, it was recognized that the
Fourier equation is in contradiction with the theory of relativity because it admits an infinite speed of propagation of heat signals. For example, according to the Fourier
369:
An example of transient conduction that does not end with steady-state conduction, but rather no conduction, occurs when a hot copper ball is dropped into oil at a low temperature. Here, the temperature field within the object begins to change as a function of time, as the heat is removed from the
349:
When a new perturbation of temperature of this type happens, temperatures within the system change in time toward a new equilibrium with the new conditions, provided that these do not change. After equilibrium, heat flow into the system once again equals the heat flow out, and temperatures at each
311:
For example, a bar may be cold at one end and hot at the other, but after a state of steady-state conduction is reached, the spatial gradient of temperatures along the bar does not change any further, as time proceeds. Instead, the temperature remains constant at any given cross-section of the rod
298:
Steady-state conduction is the form of conduction that happens when the temperature difference(s) driving the conduction are constant, so that (after an equilibration time), the spatial distribution of temperatures (temperature field) in the conducting object does not change any further. Thus, all
136:
is a measure of an interface's resistance to thermal flow. This thermal resistance differs from contact resistance, as it exists even at atomically perfect interfaces. Understanding the thermal resistance at the interface between two materials is of primary significance in the study of its thermal
406:
across their extent, during the process (as compared to the rest of the system). This is due to their far higher conductance. During transient conduction, therefore, the temperature across their conductive regions changes uniformly in space, and as a simple exponential in time. An example of such
319:. In such cases, temperature plays the role of voltage, and heat transferred per unit time (heat power) is the analog of electric current. Steady-state systems can be modeled by networks of such thermal resistances in series and parallel, in exact analogy to electrical networks of resistors. See
94:
A region with greater thermal energy (heat) corresponds with greater molecular agitation. Thus when a hot object touches a cooler surface, the highly agitated molecules from the hot object bump the calm molecules of the cooler surface, transferring the microscopic kinetic energy and causing the
64:
is heat transfer through stationary matter by physical contact. (The matter is stationary on a macroscopic scaleâwe know there is thermal motion of the atoms and molecules at any temperature above absolute zero.) Heat transferred between the electric burner of a stove and the bottom of a pan is
365:
New external conditions also cause this process: for example, the copper bar in the example steady-state conduction experiences transient conduction as soon as one end is subjected to a different temperature from the other. Over time, the field of temperatures inside the bar reaches a new
4945:
Thermal conduction property of any gas under standard conditions of pressure and temperature is a fixed quantity. This property of a known reference gas or known reference gas mixtures can, therefore, be used for certain sensory applications, such as the thermal conductivity analyzer.
29:
is the diffusion of thermal energy (heat) within one material or between materials in contact. The higher temperature object has molecules with more kinetic energy; collisions between molecules distributes this kinetic energy until an object has the same kinetic energy throughout.
863:
353:
If changes in external temperatures or internal heat generation changes are too rapid for the equilibrium of temperatures in space to take place, then the system never reaches a state of unchanging temperature distribution in time, and the system remains in a transient state.
3912:
131:
is the study of heat conduction between solid bodies in contact. A temperature drop is often observed at the interface between the two surfaces in contact. This phenomenon is said to be a result of a thermal contact resistance existing between the contacting surfaces.
38:, is a property that relates the rate of heat loss per unit area of a material to its rate of change of temperature. Essentially, it is a value that accounts for any property of the material that could change the way it conducts heat. Heat spontaneously flows along a
4966:
one set of two filaments and the reference gas on the other set of two filaments, then the
Wheatstone bridge becomes unbalanced. And the resulting net voltage output of the circuit will be correlated with the database to identify the constituents of the sample gas.
4775:
307:
are uniformly zero. In steady-state conduction, the amount of heat entering any region of an object is equal to the amount of heat coming out (if this were not so, the temperature would be rising or falling, as thermal energy was tapped or trapped in a region).
2119:
1940:
4641:
as the boundary conditions. Splat cooling rapidly ends in a steady state temperature, and is similar in form to the
Gaussian diffusion equation. The temperature profile, with respect to the position and time of this type of cooling, varies with:
78:
occurs when microwaves, infrared radiation, visible light, or another form of electromagnetic radiation is emitted or absorbed. An obvious example is the warming of the Earth by the Sun. A less obvious example is thermal radiation from the human
3162:
738:
is often treated as a constant, though this is not always true. While the thermal conductivity of a material generally varies with temperature, the variation can be small over a significant range of temperatures for some common materials. In
2112:
4259:
425:
equation, a pulse of heat at the origin would be felt at infinity instantaneously. The speed of information propagation is faster than the speed of light in vacuum, which is physically inadmissible within the framework of relativity.
3386:
382:
within the shape (i.e., most complex objects, mechanisms or machines in engineering) often the application of approximate theories is required, and/or numerical analysis by computer. One popular graphical method involves the use of
373:
The analysis of non-steady-state conduction systems is more complex than that of steady-state systems. If the conducting body has a simple shape, then exact analytical mathematical expressions and solutions may be possible (see
3034:
3274:
3494:
95:
colder part or object to heat up. Mathematically, thermal conduction works just like diffusion. As temperature difference goes up, the distance traveled gets shorter or the area goes up thermal conduction increases:
3710:
4645:
1812:
3592:
1338:
1122:
4375:
4868:(TTT). It is possible to manipulate the cooling process to adjust the phase of a suitable material. For example, appropriate quenching of steel can convert a desirable proportion of its content of
3998:
The heat transfer at an interface is considered a transient heat flow. To analyze this problem, the Biot number is important to understand how the system behaves. The Biot number is determined by:
1232:
4911:
is a physical connection between two bodies that allows the passage of heat between them. Bailyn is referring to diathermal walls that exclusively connect two bodies, especially conductive walls.
1845:
1467:
315:
In steady-state conduction, all the laws of direct current electrical conduction can be applied to "heat currents". In such cases, it is possible to take "thermal resistances" as the analog to
4312:, and represents the transfer of heat at an interface between two materials. This value is different at every interface and is an important concept in understanding heat flow at an interface.
1014:{\displaystyle {\frac {\partial T}{\partial t}}=\alpha \left({\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}}\right)}
4850:
2717:
4038:
1633:
4100:
4310:
3039:
2019:
4639:
2545:
853:
1740:
1395:
624:
2597:
2024:
1685:
4447:
is a method for quenching small droplets of molten materials by rapid contact with a cold surface. The particles undergo a characteristic cooling process, with the heat profile at
4144:
3279:
2923:
4139:
1303:
1196:
1087:
3703:
125:
vibrations of particles harder to transmit. Gases have even more space, and therefore infrequent particle collisions. This makes liquids and gases poor conductors of heat.
653:
535:
4914:
This statement of the "zeroth law" belongs to an idealized theoretical discourse, and actual physical walls may have peculiarities that do not conform to its generality.
2878:
1276:
4552:
2480:
2429:
2278:{\displaystyle {\frac {\Delta Q}{\Delta t}}={\frac {A\,(-\Delta T)}{{\frac {\Delta x_{1}}{k_{1}}}+{\frac {\Delta x_{2}}{k_{2}}}+{\frac {\Delta x_{3}}{k_{3}}}+\cdots }}.}
4604:
4578:
2389:
2353:
1840:
584:
1560:
1493:
712:
4805:
2930:
2754:
2291:
of fluid that remains stationary next to the barrier. This thin film of fluid is difficult to quantify because its characteristics depend upon complex conditions of
4425:
3654:
3627:
2838:
2818:
2791:
2663:
2631:
1146:
235:
46:
of an electric stove to the bottom of a saucepan in contact with it. In the absence of an opposing external driving energy source, within a body or between bodies,
3167:
4523:
4497:
4471:
350:
point inside the system no longer change. Once this happens, transient conduction is ended, although steady-state conduction may continue if heat flow continues.
1169:
4058:
1582:
1535:
1513:
1366:
1055:
785:
761:
736:
679:
558:
71:
is the heat transfer by the macroscopic movement of a fluid. This type of transfer takes place in a forced-air furnace and in weather systems, for example.
390:
Occasionally, transient conduction problems may be considerably simplified if regions of the object being heated or cooled can be identified, for which
4904:
is directly focused on the idea of conduction of heat. Bailyn (1994) writes that "the zeroth law may be stated: All diathermal walls are equivalent".
179:
metallic solid conducts most of the heat flux through the solid. Phonon flux is still present but carries less of the energy. Electrons also conduct
1752:
394:
is very much greater than that for heat paths leading into the region. In this case, the region with high conductivity can often be treated in the
312:
normal to the direction of heat transfer, and this temperature varies linearly in space in the case where there is no heat generation in the rod.
5183:
contains a variety of transient expressions for heat conduction, along with algorithms and computer code for obtaining precise numerical values.
456:. It is known as "second sound" because the wave motion of heat is similar to the propagation of sound in air.this is called Quantum conduction
4328:
3391:
140:
The inter-molecular transfer of energy could be primarily by elastic impact, as in fluids, or by free-electron diffusion, as in metals, or
5086:
1416:
4379:
The Biot number increases as the
Fourier number decreases. There are five steps to determine a temperature profile in terms of time.
2674:
320:
4962:
The principle of thermal conductivity of gases can also be used to measure the concentration of a gas in a binary mixture of gases.
4001:
2758:
The rules for combining resistances and conductances (in series and parallel) are the same for both heat flow and electric current.
1595:
4063:
4892:. By calculating the heat transfer coefficient from this Biot number, one can find a liquid medium suitable for the application.
1959:
3907:{\displaystyle {\dot {Q}}=4k\pi {\frac {T_{1}-T_{2}}{1/{r_{1}}-1/{r_{2}}}}=4k\pi {\frac {(T_{1}-T_{2})r_{1}r_{2}}{r_{2}-r_{1}}}}
1308:
1092:
3499:
2504:
799:
2287:
For heat conduction from one fluid to another through a barrier, it is sometimes important to consider the conductance of the
1956:
are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by:
1704:
589:
5333:
5299:
5265:
5231:
5204:
2557:
346:
source or sink of heat suddenly introduced within an object, causing temperatures near the source or sink to change in time.
290:
is a quantity derived from conductivity, which is a measure of its ability to exchange thermal energy with its surroundings.
1648:
1201:
4770:{\displaystyle T(x,t)-T_{i}={\frac {T_{i}\Delta X}{2{\sqrt {\pi \alpha t}}}}\exp \left(-{\frac {x^{2}}{4\alpha t}}\right)}
3959:
199:
is caused by the interaction of heat flux and electric current. Heat conduction within a solid is directly analogous to
5395:
2312:
2308:
1413:
for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as
5384:
Liqiu Wang, Xuesheng Zhou, Xiaohao Wei, 'Heat
Conduction: Mathematical Models and Analytical Solutions' Springer 2008
5381:
Jan Taler, Piotr Duda, 'Solving Direct and
Inverse Heat Conduction Problems' Springer-Verlag Berlin Heidelberg 2005
5364:
5156:
5039:
4865:
4810:
4105:
3985:
3967:
5418:
5400:
John H Lienhard IV and John H Lienhard V, 'A Heat
Transfer Textbook', Fifth Edition, Dover Pub., Mineola, NY, 2019
5014:
108:
Thermal conduction (power) is the heat per unit time transferred some distance â between the two temperatures.
4276:
2315:. Ideally, the formulae for conductance should produce a quantity with dimensions independent of distance, like
5424:
5378:
F. Dehghani, CHNG2801 â 'Conservation and
Transport Processes: Course Notes', University of Sydney, Sydney 2007
4609:
5438:
4979:
3963:
133:
1373:
4989:
477:
1935:{\displaystyle R={\frac {1}{U}}={\frac {\Delta x}{k}}={\frac {A\,(-\Delta T)}{\frac {\Delta Q}{\Delta t}}}.}
5469:
5464:
5009:
4901:
421:
500:
128:
20:
4876:, creating a very hard and strong product. To achieve this, it is necessary to quench at the "nose" (or
5004:
2886:
492:
408:
4264:
3157:{\displaystyle {\dot {Q}}\int _{r_{1}}^{r_{2}}{\frac {1}{r}}\,dr=-2k\pi \ell \int _{T_{1}}^{T_{2}}dT}
790:
For many simple applications, Fourier's law is used in its one-dimensional form, for example, in the
4908:
1944:
Resistance is additive when several conducting layers lie between the hot and cool regions, because
1286:
1179:
1070:
5454:
3948:
3666:
1022:
395:
283:
137:
properties. Interfaces often contribute significantly to the observed properties of the materials.
636:
518:
3952:
2843:
1249:
188:
4528:
2446:
2397:
5459:
5375:
H. S. Carslaw and J. C. Jaeger 'Conduction of heat in solids' Oxford University Press, USA 1959
5062:
4583:
4557:
2766:
Conduction through cylindrical shells (e.g. pipes) can be calculated from the internal radius,
2358:
2322:
2107:{\displaystyle {\frac {1}{U}}={\frac {1}{U_{1}}}+{\frac {1}{U_{2}}}+{\frac {1}{U_{3}}}+\cdots }
1819:
858:
563:
5323:
5289:
5255:
5221:
4779:
Splat cooling is a fundamental concept that has been adapted for practical use in the form of
1542:
1475:
694:
586:. The heat flux density is the amount of energy that flows through a unit area per unit time.
378:
for the analytical approach). However, most often, because of complicated shapes with varying
5121:
4984:
4790:
4254:{\displaystyle {\frac {T-T_{f}}{T_{i}-T_{f}}}=\exp \left({\frac {-hAt}{\rho C_{p}V}}\right).}
2299:âbut when dealing with thin high-conductance barriers it can sometimes be quite significant.
1406:
359:
316:
2722:
4994:
4403:
3632:
3605:
2823:
2796:
2769:
2636:
2602:
1825:
1127:
682:
538:
453:
391:
379:
303:
may either be zero or have nonzero values, but all derivatives of temperature at any point
249:
241:
213:
184:
176:
39:
3381:{\displaystyle R_{c}={\frac {\Delta T}{\dot {Q}}}={\frac {\ln(r_{2}/r_{1})}{2\pi k\ell }}}
8:
5430:
5196:
Science of Sintering: New Directions for Materials Processing and Microstructural Control
4784:
4502:
4476:
4450:
1642:
171:) have free-moving electrons that transfer thermal energy rapidly through the metal. The
51:
1151:
398:, as a "lump" of material with a simple thermal capacitance consisting of its aggregate
5125:
4929:
4043:
1589:
1567:
1520:
1498:
1351:
1040:
770:
746:
721:
664:
543:
437:
287:
145:
5391:
5390:
Latif M. Jiji, Amir H. Danesh-Yazdi, 'Heat Conduction' Springer, Fourth Edition 2024
5360:
5329:
5295:
5261:
5227:
5200:
5162:
5152:
5129:
5105:
4950:
2116:
So, when dealing with a multilayer partition, the following formula is usually used:
278:
196:
743:
materials, the thermal conductivity typically varies with orientation; in this case
452:. Heat takes the place of pressure in normal sound waves. This leads to a very high
42:(i.e. from a hotter body to a colder body). For example, heat is conducted from the
5117:
4918:
4780:
465:
240:
To quantify the ease with which a particular medium conducts, engineers employ the
180:
511:
The differential form of Fourier's law of thermal conduction shows that the local
5434:
5194:
5144:
5034:
5029:
4877:
203:
of particles within a fluid, in the situation where there are no fluid currents.
156:
3029:{\displaystyle {\dot {Q}}=-kA_{r}{\frac {dT}{dr}}=-2k\pi r\ell {\frac {dT}{dr}}}
5087:"5.6 Heat Transfer Methods â Conduction, Convection and Radiation Introduction"
5019:
4885:
208:
191:
of most metals have about the same ratio. A good electrical conductor, such as
164:
5448:
5166:
4925:
4444:
3922:
3707:
Solving in a similar manner as for a cylindrical shell (see above) produces:
3269:{\displaystyle {\dot {Q}}=2k\pi \ell {\frac {T_{1}-T_{2}}{\ln(r_{2}/r_{1})}}}
1690:
1342:
473:
469:
445:
441:
399:
384:
375:
332:
160:
89:
5401:
5287:
3660:
2316:
1638:
496:
433:
207:
path of gas molecules relative to the size of the gas gap, as given by the
4430:
Compared required to point to trace specified Biot number on the nomogram.
3489:{\textstyle {\dot {Q}}=2\pi k\ell r_{m}{\frac {T_{1}-T_{2}}{r_{2}-r_{1}}}}
5024:
4881:
1398:
1026:
341:
at any place within an object, the mode of thermal energy flow is termed
168:
57:
Every process involving heat transfer takes place by only three methods:
47:
5387:
W. Kelly, 'Understanding Heat Conduction' Nova Science Publischer, 2010
5106:"Effective Thermal Conductivity of Submicron Powders: A Numerical Study"
1637:
With a simple 1-D steady heat conduction equation which is analogous to
4873:
2292:
1037:
By integrating the differential form over the material's total surface
740:
159:
of thermal energy. This is due to the way that metals bond chemically:
5193:
III, H. Palmour; Spriggs, R. M.; Uskokovic, D. P. (11 November 2013).
4999:
4869:
4861:
2296:
2288:
1345:
transferred by conduction (in W), time derivative of the transferred
512:
486:
449:
244:, also known as the conductivity constant or conduction coefficient,
200:
3937:
3656:, can be calculated in a similar manner as for a cylindrical shell.
4889:
4316:
2840:, and the temperature difference between the inner and outer wall,
1410:
481:
43:
5288:
Rajiv Asthana; Ashok Kumar; Narendra B. Dahotre (9 January 2006).
402:. Such regions warm or cool, but show no significant temperature
5413:
3602:
The conduction through a spherical shell with internal radius,
764:
686:
192:
149:
141:
117:ÎT is the difference in temperature from one side to the other.
1807:{\displaystyle {\frac {\Delta Q}{\Delta t}}=UA\,(-\Delta T).}
155:
Metals (e.g., copper, platinum, gold, etc.) are usually good
5142:
4954:
be correlated with the database to identify the gas sample.
4949:
The working of this instrument is by principle based on the
4924:
These differences are among the defining characteristics of
3594:. It is important to note that this is the log-mean radius.
1346:
656:
120:â is the length of the path the heat has to be transferred.
4325:
coordinate and the Fourier number, which is calculated by
3587:{\textstyle r_{m}={\frac {r_{2}-r_{1}}{\ln(r_{2}/r_{1})}}}
1333:{\displaystyle \mathbf {q} \cdot \mathrm {d} \mathbf {S} }
1117:{\displaystyle \mathbf {q} \cdot \mathrm {d} \mathbf {S} }
5281:
4917:
For example, the material of the wall must not undergo a
4370:{\displaystyle {\textit {Fo}}={\frac {\alpha t}{L^{2}}}.}
448:-like motion, rather than by the more usual mechanism of
2307:
The previous conductance equations, written in terms of
5219:
5180:
4884:, the time it takes for the material to quench, or the
1239:
627:
5199:. Springer Science & Business Media. p. 164.
4880:) of the TTT diagram. Since materials differ in their
3502:
3394:
1290:
1227:{\displaystyle \nabla T\cdot \mathrm {d} \mathbf {S} }
1183:
1074:
4864:
is a transient heat transfer process in terms of the
4813:
4793:
4648:
4612:
4586:
4560:
4531:
4505:
4479:
4453:
4427:
to relative temperature with the boundary conditions.
4406:
4331:
4279:
4147:
4108:
4066:
4046:
4004:
3713:
3669:
3635:
3608:
3282:
3170:
3042:
2933:
2889:
2846:
2826:
2799:
2772:
2725:
2677:
2639:
2605:
2560:
2507:
2449:
2400:
2361:
2325:
2122:
2027:
1962:
1848:
1822:
1755:
1707:
1651:
1598:
1570:
1545:
1523:
1501:
1495:
is the time interval during which the amount of heat
1478:
1419:
1376:
1354:
1311:
1289:
1252:
1204:
1182:
1154:
1130:
1095:
1073:
1043:
866:
802:
773:
749:
724:
697:
667:
639:
592:
566:
546:
521:
216:
16:
Process by which heat is transferred within an object
5325:
Handbook of Residual Stress and Deformation of Steel
4060:, is introduced in this formula, and is measured in
2302:
5192:
1689:This law forms the basis for the derivation of the
1462:{\displaystyle Q=-k{\frac {A\Delta t}{L}}\Delta T,}
1057:, we arrive at the integral form of Fourier's law:
857:In an isotropic medium, Fourier's law leads to the
4844:
4799:
4769:
4633:
4598:
4572:
4546:
4517:
4491:
4465:
4419:
4369:
4304:
4253:
4133:
4094:
4052:
4032:
3906:
3697:
3648:
3621:
3586:
3488:
3380:
3268:
3156:
3028:
2917:
2872:
2832:
2812:
2785:
2748:
2711:
2657:
2625:
2591:
2539:
2474:
2423:
2383:
2347:
2277:
2106:
2013:
1934:
1834:
1806:
1734:
1679:
1627:
1576:
1554:
1529:
1507:
1487:
1461:
1389:
1360:
1332:
1297:
1270:
1226:
1190:
1163:
1140:
1116:
1081:
1049:
1013:
847:
779:
755:
730:
706:
673:
647:
618:
578:
552:
529:
229:
54:is approached, temperature becoming more uniform.
2471:
2380:
2344:
5446:
5321:
5147:; Incropera, Frank P.; Dewitt, David P. (2011).
4940:
499:is the electrical analogue of Fourier's law and
337:During any period in which temperatures changes
4895:
4845:{\displaystyle \alpha ={\frac {k}{\rho C_{p}}}}
4386:Determine which relative depth matters, either
4319:. A nomogram has a relative temperature as the
3916:
2712:{\displaystyle R={\frac {\Delta T}{\dot {Q}}},}
1562:is the temperature difference between the ends,
495:is a discrete analogue of Fourier's law, while
286:, temperature, density, and molecular bonding.
268:), in a direction normal to a surface of area (
5291:Materials Processing and Manufacturing Science
5220:Sam Zhang; Dongliang Zhao (19 November 2012).
4033:{\displaystyle {\textit {Bi}}={\frac {hL}{k}}}
1628:{\displaystyle R={\frac {1}{k}}{\frac {L}{A}}}
1515:flows through a cross-section of the material,
148:, the heat flux is carried almost entirely by
5223:Aeronautical and Aerospace Materials Handbook
4095:{\displaystyle \mathrm {\frac {J}{m^{2}sK}} }
2667:The reciprocal of conductance is resistance,
1816:The reciprocal of conductance is resistance,
111:Îș is the thermal conductivity of the material
5213:
4305:{\displaystyle \mathrm {\frac {W}{m^{2}K}} }
560:and the negative local temperature gradient
282:that is primarily dependent on the medium's
5359:, American Institute of Physics, New York,
5253:
4315:The series solution can be analyzed with a
3966:. Unsourced material may be challenged and
2014:{\displaystyle R=R_{1}+R_{2}+R_{3}+\cdots }
114:A is the cross-sectional area of the object
5315:
4634:{\displaystyle -\infty \leq x\leq \infty }
4473:for initial temperature as the maximum at
3927:
464:The law of heat conduction, also known as
415:
293:
4852:. This varies according to the material.
4121:
3986:Learn how and when to remove this message
3094:
2579:
2540:{\displaystyle U={\frac {kA}{\Delta x}},}
2470:
2379:
2343:
2152:
1892:
1785:
1664:
848:{\displaystyle q_{x}=-k{\frac {dT}{dx}}.}
5294:. ButterworthâHeinemann. pp. 158â.
1735:{\displaystyle U={\frac {k}{\Delta x}},}
1390:{\displaystyle \mathrm {d} \mathbf {S} }
619:{\displaystyle \mathbf {q} =-k\nabla T,}
276:) ". Thermal conductivity is a material
5247:
2592:{\displaystyle {\dot {Q}}=U\,\Delta T,}
326:
5447:
5149:Fundamentals of heat and mass transfer
5122:10.4028/www.scientific.net/AMM.846.500
1680:{\displaystyle \Delta T=R\,{\dot {Q}}}
5433:by Jeff Bryant based on a program by
4334:
4007:
2761:
2554:Fourier's law can also be stated as:
1749:Fourier's law can also be stated as:
428:
272:), due to a temperature difference (Î
256:is defined as "the quantity of heat,
5328:. ASM International. pp. 322â.
5181:Exact Analytical Conduction Toolbox
5151:(7th ed.). Hoboken, NJ: Wiley.
5057:
5055:
3964:adding citations to reliable sources
3931:
2927:When Fourier's equation is applied:
2883:The surface area of the cylinder is
1537:is the cross-sectional surface area,
506:
5103:
4397:Convert time to the Fourier number.
3164:then the rate of heat transfer is:
299:partial derivatives of temperature
183:through conductive solids, and the
13:
4855:
4696:
4628:
4616:
4593:
4567:
4541:
4296:
4287:
4282:
4122:
4086:
4083:
4074:
4069:
3299:
2687:
2580:
2525:
2311:, can be reformulated in terms of
2237:
2205:
2173:
2159:
2134:
2126:
1919:
1911:
1899:
1871:
1792:
1767:
1759:
1720:
1652:
1546:
1479:
1450:
1438:
1378:
1321:
1215:
1205:
1105:
990:
976:
953:
939:
916:
902:
878:
870:
698:
607:
570:
323:for an example of such a network.
14:
5481:
5406:
5052:
5040:General equation of heat transfer
2918:{\displaystyle A_{r}=2\pi r\ell }
2443:is cross-sectional area, we have
2303:Intensive-property representation
1592:of the 1-D homogeneous material:
1588:One can define the (macroscopic)
1584:is the distance between the ends.
763:is represented by a second-order
714:is the temperature gradient, K/m.
321:purely resistive thermal circuits
50:differences decay over time, and
5412:
4439:
3936:
1746:is the conductance, in W/(m K).
1383:
1326:
1313:
1278:
1220:
1171:
1110:
1097:
1062:
1032:
655:is the local heat flux density,
641:
594:
523:
459:
5349:
5110:Applied Mechanics and Materials
4900:One statement of the so-called
4866:time temperature transformation
4434:
4134:{\displaystyle q=-h\,\Delta T,}
5439:Wolfram Demonstrations Project
5186:
5173:
5136:
5097:
5079:
4980:List of thermal conductivities
4935:
4664:
4652:
4040:The heat transfer coefficient
3853:
3827:
3578:
3550:
3358:
3330:
3260:
3232:
2165:
2153:
1905:
1893:
1798:
1786:
1696:
1298:{\displaystyle \scriptstyle S}
1191:{\displaystyle \scriptstyle S}
1082:{\displaystyle \scriptstyle S}
787:varies with spatial location.
134:Interfacial thermal resistance
1:
5045:
4990:Convection diffusion equation
4957:
4941:Thermal conductivity analyzer
3698:{\displaystyle A=4\pi r^{2}.}
2394:From the electrical formula:
1399:oriented surface area element
407:systems is those that follow
5226:. CRC Press. pp. 304â.
5015:ChurchillâBernstein equation
5010:Relativistic heat conduction
4902:zeroth law of thermodynamics
4896:Zeroth law of thermodynamics
4787:coefficient, represented as
3917:Transient thermal conduction
3597:
767:. In non-uniform materials,
648:{\displaystyle \mathbf {q} }
530:{\displaystyle \mathbf {q} }
422:relativistic heat conduction
34:, frequently represented by
7:
5260:. Springer. pp. 174â.
4973:
3276:the thermal resistance is:
2873:{\displaystyle T_{2}-T_{1}}
2319:for electrical resistance,
1271:{\displaystyle {\dot {Q}}=}
537:is equal to the product of
472:), states that the rate of
195:, also conducts heat well.
129:Thermal contact conductance
99:Thermal conduction (power)=
83:
21:Conduction (disambiguation)
10:
5488:
5357:A Survey of Thermodynamics
4580:, and the heat profile at
4547:{\displaystyle x=-\infty }
3920:
2475:{\displaystyle G=kA/x\,\!}
2424:{\displaystyle R=\rho x/A}
503:is its chemical analogue.
330:
87:
65:transferred by conduction.
18:
5322:George E. Totten (2002).
5257:Drop-Surface Interactions
5104:Dai; et al. (2015).
4599:{\displaystyle t=\infty }
4573:{\displaystyle x=\infty }
4383:Calculate the Biot number
4265:heat transfer coefficient
2498:is cross-sectional area.
2384:{\displaystyle G=I/V\,\!}
2348:{\displaystyle R=V/I\,\!}
1835:{\displaystyle {\big .}R}
718:The thermal conductivity
579:{\displaystyle -\nabla T}
189:electrical conductivities
2719:analogous to Ohm's law,
2599:analogous to Ohm's law,
1555:{\displaystyle \Delta T}
1488:{\displaystyle \Delta t}
707:{\displaystyle \nabla T}
501:Fick's laws of diffusion
396:lumped capacitance model
5431:Newton's Law of Cooling
5091:Douglas College Physics
5005:Fick's law of diffusion
4928:. In a sense, they are
4800:{\displaystyle \alpha }
3928:Interface heat transfer
3629:, and external radius,
2793:, the external radius,
493:Newton's law of cooling
416:Relativistic conduction
409:Newton's law of cooling
294:Steady-state conduction
264:) through a thickness (
260:, transmitted in time (
144:, as in insulators. In
5143:Bergman, Theodore L.;
4846:
4801:
4771:
4635:
4600:
4574:
4548:
4519:
4493:
4467:
4421:
4371:
4306:
4255:
4135:
4096:
4054:
4034:
3908:
3699:
3650:
3623:
3588:
3490:
3382:
3270:
3158:
3030:
2919:
2874:
2834:
2814:
2787:
2750:
2749:{\displaystyle R=V/I.}
2713:
2659:
2627:
2593:
2541:
2476:
2425:
2385:
2349:
2279:
2108:
2015:
1936:
1836:
1808:
1736:
1681:
1629:
1578:
1556:
1531:
1509:
1489:
1463:
1391:
1362:
1334:
1299:
1272:
1228:
1192:
1165:
1142:
1118:
1083:
1051:
1025:famously known as the
1015:
849:
781:
757:
732:
708:
675:
649:
620:
580:
554:
531:
476:through a material is
380:thermal conductivities
317:electrical resistances
231:
5427:â Thermal-FluidsPedia
4985:Electrical conduction
4970:to that of nitrogen.
4847:
4802:
4772:
4636:
4601:
4575:
4549:
4520:
4494:
4468:
4422:
4420:{\displaystyle T_{i}}
4372:
4307:
4256:
4136:
4097:
4055:
4035:
3909:
3700:
3651:
3649:{\displaystyle r_{2}}
3624:
3622:{\displaystyle r_{1}}
3589:
3491:
3383:
3271:
3159:
3031:
2920:
2875:
2835:
2833:{\displaystyle \ell }
2815:
2813:{\displaystyle r_{2}}
2788:
2786:{\displaystyle r_{1}}
2751:
2714:
2660:
2658:{\displaystyle I=VG.}
2628:
2626:{\displaystyle I=V/R}
2594:
2542:
2477:
2426:
2386:
2350:
2280:
2109:
2016:
1937:
1837:
1809:
1737:
1682:
1630:
1579:
1557:
1532:
1510:
1490:
1464:
1407:differential equation
1392:
1363:
1335:
1300:
1273:
1238:where (including the
1229:
1193:
1166:
1143:
1141:{\displaystyle {}={}}
1119:
1084:
1052:
1016:
850:
782:
758:
733:
709:
676:
650:
626:where (including the
621:
581:
555:
532:
360:operating temperature
343:transient conduction.
232:
230:{\displaystyle K_{n}}
5421:at Wikimedia Commons
5254:Martin Eein (2002).
4995:R-value (insulation)
4811:
4807:, can be written as
4791:
4646:
4610:
4584:
4558:
4529:
4503:
4477:
4451:
4404:
4329:
4277:
4145:
4106:
4064:
4044:
4002:
3960:improve this section
3711:
3667:
3633:
3606:
3500:
3392:
3280:
3168:
3040:
2931:
2887:
2844:
2824:
2797:
2770:
2723:
2675:
2637:
2603:
2558:
2551:is the conductance.
2505:
2447:
2398:
2359:
2323:
2313:intensive properties
2309:extensive properties
2120:
2025:
1960:
1846:
1820:
1753:
1705:
1649:
1596:
1568:
1543:
1521:
1499:
1476:
1417:
1374:
1352:
1309:
1287:
1250:
1202:
1180:
1152:
1128:
1093:
1071:
1041:
1023:fundamental solution
864:
800:
771:
747:
722:
695:
665:
637:
590:
564:
544:
539:thermal conductivity
519:
454:thermal conductivity
440:phenomenon in which
392:thermal conductivity
327:Transient conduction
250:thermal conductivity
242:thermal conductivity
214:
40:temperature gradient
32:Thermal conductivity
19:For other uses, see
5470:Transport phenomena
5465:Physical quantities
5355:Bailyn, M. (1994).
5145:Lavine, Adrienne S.
4785:thermal diffusivity
4518:{\displaystyle T=0}
4492:{\displaystyle x=0}
4466:{\displaystyle t=0}
3147:
3083:
2355:, and conductance,
1643:electric resistance
489:of energy locally.
468:(compare Fourier's
52:thermal equilibrium
5067:energyeducation.ca
5063:"Energy Education"
4932:of heat transfer.
4842:
4797:
4767:
4631:
4596:
4570:
4544:
4515:
4489:
4463:
4417:
4367:
4302:
4251:
4131:
4092:
4050:
4030:
3904:
3695:
3663:of the sphere is:
3646:
3619:
3584:
3486:
3378:
3266:
3154:
3119:
3055:
3026:
2915:
2870:
2830:
2810:
2783:
2762:Cylindrical shells
2746:
2709:
2655:
2623:
2589:
2537:
2472:
2421:
2381:
2345:
2275:
2104:
2011:
1932:
1832:
1804:
1732:
1677:
1625:
1590:thermal resistance
1574:
1552:
1527:
1505:
1485:
1459:
1387:
1358:
1330:
1295:
1294:
1268:
1224:
1188:
1187:
1164:{\displaystyle -k}
1161:
1138:
1114:
1079:
1078:
1047:
1011:
845:
777:
753:
728:
704:
681:is the material's
671:
645:
616:
576:
550:
527:
438:quantum mechanical
429:Quantum conduction
288:Thermal effusivity
227:
27:Thermal conduction
5417:Media related to
5335:978-1-61503-227-3
5301:978-0-08-046488-6
5267:978-3-211-83692-7
5233:978-1-4398-7329-8
5206:978-1-4899-0933-6
4951:Wheatstone bridge
4840:
4760:
4721:
4718:
4362:
4336:
4300:
4273:, is measured in
4242:
4193:
4090:
4053:{\displaystyle h}
4028:
4009:
3996:
3995:
3988:
3902:
3810:
3723:
3582:
3484:
3404:
3376:
3316:
3314:
3264:
3180:
3092:
3052:
3024:
2983:
2943:
2704:
2702:
2570:
2532:
2490:is conductivity,
2270:
2261:
2229:
2197:
2141:
2096:
2076:
2056:
2036:
1927:
1926:
1881:
1863:
1774:
1727:
1674:
1623:
1613:
1577:{\displaystyle L}
1530:{\displaystyle A}
1508:{\displaystyle Q}
1448:
1361:{\displaystyle Q}
1262:
1050:{\displaystyle S}
1004:
967:
930:
885:
840:
780:{\displaystyle k}
756:{\displaystyle k}
731:{\displaystyle k}
674:{\displaystyle k}
553:{\displaystyle k}
507:Differential form
197:Thermoelectricity
74:Heat transfer by
5477:
5416:
5368:
5353:
5347:
5346:
5344:
5342:
5319:
5313:
5312:
5310:
5308:
5285:
5279:
5278:
5276:
5274:
5251:
5245:
5244:
5242:
5240:
5217:
5211:
5210:
5190:
5184:
5177:
5171:
5170:
5140:
5134:
5133:
5101:
5095:
5094:
5083:
5077:
5076:
5074:
5073:
5059:
4919:phase transition
4851:
4849:
4848:
4843:
4841:
4839:
4838:
4837:
4821:
4806:
4804:
4803:
4798:
4781:thermal spraying
4776:
4774:
4773:
4768:
4766:
4762:
4761:
4759:
4748:
4747:
4738:
4722:
4720:
4719:
4708:
4702:
4695:
4694:
4684:
4679:
4678:
4640:
4638:
4637:
4632:
4605:
4603:
4602:
4597:
4579:
4577:
4576:
4571:
4553:
4551:
4550:
4545:
4524:
4522:
4521:
4516:
4498:
4496:
4495:
4490:
4472:
4470:
4469:
4464:
4426:
4424:
4423:
4418:
4416:
4415:
4376:
4374:
4373:
4368:
4363:
4361:
4360:
4351:
4343:
4338:
4337:
4324:
4311:
4309:
4308:
4303:
4301:
4299:
4295:
4294:
4281:
4272:
4260:
4258:
4257:
4252:
4247:
4243:
4241:
4237:
4236:
4223:
4209:
4194:
4192:
4191:
4190:
4178:
4177:
4167:
4166:
4165:
4149:
4140:
4138:
4137:
4132:
4101:
4099:
4098:
4093:
4091:
4089:
4082:
4081:
4068:
4059:
4057:
4056:
4051:
4039:
4037:
4036:
4031:
4029:
4024:
4016:
4011:
4010:
3991:
3984:
3980:
3977:
3971:
3940:
3932:
3913:
3911:
3910:
3905:
3903:
3901:
3900:
3899:
3887:
3886:
3876:
3875:
3874:
3865:
3864:
3852:
3851:
3839:
3838:
3825:
3811:
3809:
3808:
3807:
3806:
3796:
3785:
3784:
3783:
3773:
3764:
3763:
3762:
3750:
3749:
3739:
3725:
3724:
3716:
3704:
3702:
3701:
3696:
3691:
3690:
3655:
3653:
3652:
3647:
3645:
3644:
3628:
3626:
3625:
3620:
3618:
3617:
3593:
3591:
3590:
3585:
3583:
3581:
3577:
3576:
3567:
3562:
3561:
3542:
3541:
3540:
3528:
3527:
3517:
3512:
3511:
3495:
3493:
3492:
3487:
3485:
3483:
3482:
3481:
3469:
3468:
3458:
3457:
3456:
3444:
3443:
3433:
3431:
3430:
3406:
3405:
3397:
3387:
3385:
3384:
3379:
3377:
3375:
3361:
3357:
3356:
3347:
3342:
3341:
3322:
3317:
3315:
3307:
3305:
3297:
3292:
3291:
3275:
3273:
3272:
3267:
3265:
3263:
3259:
3258:
3249:
3244:
3243:
3224:
3223:
3222:
3210:
3209:
3199:
3182:
3181:
3173:
3163:
3161:
3160:
3155:
3146:
3145:
3144:
3134:
3133:
3132:
3093:
3085:
3082:
3081:
3080:
3070:
3069:
3068:
3054:
3053:
3045:
3036:and rearranged:
3035:
3033:
3032:
3027:
3025:
3023:
3015:
3007:
2984:
2982:
2974:
2966:
2964:
2963:
2945:
2944:
2936:
2924:
2922:
2921:
2916:
2899:
2898:
2879:
2877:
2876:
2871:
2869:
2868:
2856:
2855:
2839:
2837:
2836:
2831:
2819:
2817:
2816:
2811:
2809:
2808:
2792:
2790:
2789:
2784:
2782:
2781:
2755:
2753:
2752:
2747:
2739:
2718:
2716:
2715:
2710:
2705:
2703:
2695:
2693:
2685:
2664:
2662:
2661:
2656:
2632:
2630:
2629:
2624:
2619:
2598:
2596:
2595:
2590:
2572:
2571:
2563:
2550:
2546:
2544:
2543:
2538:
2533:
2531:
2523:
2515:
2486:is conductance,
2481:
2479:
2478:
2473:
2466:
2435:is resistivity,
2430:
2428:
2427:
2422:
2417:
2390:
2388:
2387:
2382:
2375:
2354:
2352:
2351:
2346:
2339:
2284:
2282:
2281:
2276:
2271:
2269:
2262:
2260:
2259:
2250:
2249:
2248:
2235:
2230:
2228:
2227:
2218:
2217:
2216:
2203:
2198:
2196:
2195:
2186:
2185:
2184:
2171:
2168:
2147:
2142:
2140:
2132:
2124:
2113:
2111:
2110:
2105:
2097:
2095:
2094:
2082:
2077:
2075:
2074:
2062:
2057:
2055:
2054:
2042:
2037:
2029:
2021:or equivalently
2020:
2018:
2017:
2012:
2004:
2003:
1991:
1990:
1978:
1977:
1955:
1949:
1941:
1939:
1938:
1933:
1928:
1925:
1917:
1909:
1908:
1887:
1882:
1877:
1869:
1864:
1856:
1841:
1839:
1838:
1833:
1828:
1827:
1813:
1811:
1810:
1805:
1775:
1773:
1765:
1757:
1745:
1741:
1739:
1738:
1733:
1728:
1726:
1715:
1686:
1684:
1683:
1678:
1676:
1675:
1667:
1634:
1632:
1631:
1626:
1624:
1616:
1614:
1606:
1583:
1581:
1580:
1575:
1561:
1559:
1558:
1553:
1536:
1534:
1533:
1528:
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1468:
1466:
1465:
1460:
1449:
1444:
1433:
1396:
1394:
1393:
1388:
1386:
1381:
1367:
1365:
1364:
1359:
1340:
1339:
1337:
1336:
1331:
1329:
1324:
1316:
1305:
1304:
1302:
1301:
1296:
1282:
1281:
1277:
1275:
1274:
1269:
1264:
1263:
1255:
1234:
1233:
1231:
1230:
1225:
1223:
1218:
1198:
1197:
1195:
1194:
1189:
1175:
1174:
1170:
1168:
1167:
1162:
1147:
1145:
1144:
1139:
1137:
1132:
1124:
1123:
1121:
1120:
1115:
1113:
1108:
1100:
1089:
1088:
1086:
1085:
1080:
1066:
1065:
1056:
1054:
1053:
1048:
1020:
1018:
1017:
1012:
1010:
1006:
1005:
1003:
1002:
1001:
988:
984:
983:
973:
968:
966:
965:
964:
951:
947:
946:
936:
931:
929:
928:
927:
914:
910:
909:
899:
886:
884:
876:
868:
854:
852:
851:
846:
841:
839:
831:
823:
812:
811:
795:
786:
784:
783:
778:
762:
760:
759:
754:
737:
735:
734:
729:
713:
711:
710:
705:
680:
678:
677:
672:
654:
652:
651:
646:
644:
625:
623:
622:
617:
597:
585:
583:
582:
577:
559:
557:
556:
551:
536:
534:
533:
528:
526:
480:to the negative
301:concerning space
236:
234:
233:
228:
226:
225:
181:electric current
142:phonon vibration
37:
5487:
5486:
5480:
5479:
5478:
5476:
5475:
5474:
5455:Heat conduction
5445:
5444:
5435:Stephen Wolfram
5425:Heat conduction
5419:Heat conduction
5409:
5372:
5371:
5354:
5350:
5340:
5338:
5336:
5320:
5316:
5306:
5304:
5302:
5286:
5282:
5272:
5270:
5268:
5252:
5248:
5238:
5236:
5234:
5218:
5214:
5207:
5191:
5187:
5178:
5174:
5159:
5141:
5137:
5102:
5098:
5085:
5084:
5080:
5071:
5069:
5061:
5060:
5053:
5048:
5035:Heat Conduction
5030:False diffusion
4976:
4960:
4943:
4938:
4909:diathermal wall
4898:
4858:
4856:Metal quenching
4833:
4829:
4825:
4820:
4812:
4809:
4808:
4792:
4789:
4788:
4749:
4743:
4739:
4737:
4733:
4729:
4707:
4703:
4690:
4686:
4685:
4683:
4674:
4670:
4647:
4644:
4643:
4611:
4608:
4607:
4585:
4582:
4581:
4559:
4556:
4555:
4530:
4527:
4526:
4504:
4501:
4500:
4478:
4475:
4474:
4452:
4449:
4448:
4442:
4437:
4411:
4407:
4405:
4402:
4401:
4356:
4352:
4344:
4342:
4333:
4332:
4330:
4327:
4326:
4320:
4290:
4286:
4285:
4280:
4278:
4275:
4274:
4268:
4232:
4228:
4224:
4210:
4208:
4204:
4186:
4182:
4173:
4169:
4168:
4161:
4157:
4150:
4148:
4146:
4143:
4142:
4107:
4104:
4103:
4077:
4073:
4072:
4067:
4065:
4062:
4061:
4045:
4042:
4041:
4017:
4015:
4006:
4005:
4003:
4000:
3999:
3992:
3981:
3975:
3972:
3957:
3941:
3930:
3925:
3919:
3895:
3891:
3882:
3878:
3877:
3870:
3866:
3860:
3856:
3847:
3843:
3834:
3830:
3826:
3824:
3802:
3798:
3797:
3792:
3779:
3775:
3774:
3769:
3765:
3758:
3754:
3745:
3741:
3740:
3738:
3715:
3714:
3712:
3709:
3708:
3686:
3682:
3668:
3665:
3664:
3640:
3636:
3634:
3631:
3630:
3613:
3609:
3607:
3604:
3603:
3600:
3572:
3568:
3563:
3557:
3553:
3543:
3536:
3532:
3523:
3519:
3518:
3516:
3507:
3503:
3501:
3498:
3497:
3477:
3473:
3464:
3460:
3459:
3452:
3448:
3439:
3435:
3434:
3432:
3426:
3422:
3396:
3395:
3393:
3390:
3389:
3362:
3352:
3348:
3343:
3337:
3333:
3323:
3321:
3306:
3298:
3296:
3287:
3283:
3281:
3278:
3277:
3254:
3250:
3245:
3239:
3235:
3225:
3218:
3214:
3205:
3201:
3200:
3198:
3172:
3171:
3169:
3166:
3165:
3140:
3136:
3135:
3128:
3124:
3123:
3084:
3076:
3072:
3071:
3064:
3060:
3059:
3044:
3043:
3041:
3038:
3037:
3016:
3008:
3006:
2975:
2967:
2965:
2959:
2955:
2935:
2934:
2932:
2929:
2928:
2894:
2890:
2888:
2885:
2884:
2864:
2860:
2851:
2847:
2845:
2842:
2841:
2825:
2822:
2821:
2804:
2800:
2798:
2795:
2794:
2777:
2773:
2771:
2768:
2767:
2764:
2735:
2724:
2721:
2720:
2694:
2686:
2684:
2676:
2673:
2672:
2638:
2635:
2634:
2615:
2604:
2601:
2600:
2562:
2561:
2559:
2556:
2555:
2548:
2524:
2516:
2514:
2506:
2503:
2502:
2494:is length, and
2462:
2448:
2445:
2444:
2439:is length, and
2413:
2399:
2396:
2395:
2371:
2360:
2357:
2356:
2335:
2324:
2321:
2320:
2305:
2255:
2251:
2244:
2240:
2236:
2234:
2223:
2219:
2212:
2208:
2204:
2202:
2191:
2187:
2180:
2176:
2172:
2170:
2169:
2148:
2146:
2133:
2125:
2123:
2121:
2118:
2117:
2090:
2086:
2081:
2070:
2066:
2061:
2050:
2046:
2041:
2028:
2026:
2023:
2022:
1999:
1995:
1986:
1982:
1973:
1969:
1961:
1958:
1957:
1951:
1945:
1918:
1910:
1888:
1886:
1870:
1868:
1855:
1847:
1844:
1843:
1824:
1823:
1821:
1818:
1817:
1766:
1758:
1756:
1754:
1751:
1750:
1743:
1719:
1714:
1706:
1703:
1702:
1699:
1666:
1665:
1650:
1647:
1646:
1615:
1605:
1597:
1594:
1593:
1569:
1566:
1565:
1544:
1541:
1540:
1522:
1519:
1518:
1500:
1497:
1496:
1477:
1474:
1473:
1434:
1432:
1418:
1415:
1414:
1382:
1377:
1375:
1372:
1371:
1353:
1350:
1349:
1325:
1320:
1312:
1310:
1307:
1306:
1288:
1285:
1284:
1283:
1279:
1254:
1253:
1251:
1248:
1247:
1246:
1219:
1214:
1203:
1200:
1199:
1181:
1178:
1177:
1176:
1172:
1153:
1150:
1149:
1148:
1136:
1131:
1129:
1126:
1125:
1109:
1104:
1096:
1094:
1091:
1090:
1072:
1069:
1068:
1067:
1063:
1061:
1042:
1039:
1038:
1035:
997:
993:
989:
979:
975:
974:
972:
960:
956:
952:
942:
938:
937:
935:
923:
919:
915:
905:
901:
900:
898:
897:
893:
877:
869:
867:
865:
862:
861:
832:
824:
822:
807:
803:
801:
798:
797:
791:
772:
769:
768:
748:
745:
744:
723:
720:
719:
696:
693:
692:
666:
663:
662:
640:
638:
635:
634:
593:
591:
588:
587:
565:
562:
561:
545:
542:
541:
522:
520:
517:
516:
509:
462:
431:
418:
335:
329:
305:concerning time
296:
221:
217:
215:
212:
211:
163:(as opposed to
102:
100:
92:
86:
35:
24:
17:
12:
11:
5:
5485:
5484:
5473:
5472:
5467:
5462:
5457:
5443:
5442:
5428:
5422:
5408:
5407:External links
5405:
5404:
5403:
5398:
5396:978-3031437397
5388:
5385:
5382:
5379:
5376:
5370:
5369:
5348:
5334:
5314:
5300:
5280:
5266:
5246:
5232:
5212:
5205:
5185:
5172:
5157:
5135:
5096:
5078:
5050:
5049:
5047:
5044:
5043:
5042:
5037:
5032:
5027:
5022:
5020:Fourier number
5017:
5012:
5007:
5002:
4997:
4992:
4987:
4982:
4975:
4972:
4959:
4956:
4942:
4939:
4937:
4934:
4897:
4894:
4886:Fourier number
4857:
4854:
4836:
4832:
4828:
4824:
4819:
4816:
4796:
4765:
4758:
4755:
4752:
4746:
4742:
4736:
4732:
4728:
4725:
4717:
4714:
4711:
4706:
4701:
4698:
4693:
4689:
4682:
4677:
4673:
4669:
4666:
4663:
4660:
4657:
4654:
4651:
4630:
4627:
4624:
4621:
4618:
4615:
4595:
4592:
4589:
4569:
4566:
4563:
4543:
4540:
4537:
4534:
4514:
4511:
4508:
4488:
4485:
4482:
4462:
4459:
4456:
4441:
4438:
4436:
4433:
4432:
4431:
4428:
4414:
4410:
4398:
4395:
4384:
4366:
4359:
4355:
4350:
4347:
4341:
4298:
4293:
4289:
4284:
4250:
4246:
4240:
4235:
4231:
4227:
4222:
4219:
4216:
4213:
4207:
4203:
4200:
4197:
4189:
4185:
4181:
4176:
4172:
4164:
4160:
4156:
4153:
4141:which becomes
4130:
4127:
4124:
4120:
4117:
4114:
4111:
4088:
4085:
4080:
4076:
4071:
4049:
4027:
4023:
4020:
4014:
3994:
3993:
3944:
3942:
3935:
3929:
3926:
3921:Main article:
3918:
3915:
3898:
3894:
3890:
3885:
3881:
3873:
3869:
3863:
3859:
3855:
3850:
3846:
3842:
3837:
3833:
3829:
3823:
3820:
3817:
3814:
3805:
3801:
3795:
3791:
3788:
3782:
3778:
3772:
3768:
3761:
3757:
3753:
3748:
3744:
3737:
3734:
3731:
3728:
3722:
3719:
3694:
3689:
3685:
3681:
3678:
3675:
3672:
3643:
3639:
3616:
3612:
3599:
3596:
3580:
3575:
3571:
3566:
3560:
3556:
3552:
3549:
3546:
3539:
3535:
3531:
3526:
3522:
3515:
3510:
3506:
3480:
3476:
3472:
3467:
3463:
3455:
3451:
3447:
3442:
3438:
3429:
3425:
3421:
3418:
3415:
3412:
3409:
3403:
3400:
3374:
3371:
3368:
3365:
3360:
3355:
3351:
3346:
3340:
3336:
3332:
3329:
3326:
3320:
3313:
3310:
3304:
3301:
3295:
3290:
3286:
3262:
3257:
3253:
3248:
3242:
3238:
3234:
3231:
3228:
3221:
3217:
3213:
3208:
3204:
3197:
3194:
3191:
3188:
3185:
3179:
3176:
3153:
3150:
3143:
3139:
3131:
3127:
3122:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3091:
3088:
3079:
3075:
3067:
3063:
3058:
3051:
3048:
3022:
3019:
3014:
3011:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2981:
2978:
2973:
2970:
2962:
2958:
2954:
2951:
2948:
2942:
2939:
2914:
2911:
2908:
2905:
2902:
2897:
2893:
2867:
2863:
2859:
2854:
2850:
2829:
2820:, the length,
2807:
2803:
2780:
2776:
2763:
2760:
2745:
2742:
2738:
2734:
2731:
2728:
2708:
2701:
2698:
2692:
2689:
2683:
2680:
2654:
2651:
2648:
2645:
2642:
2622:
2618:
2614:
2611:
2608:
2588:
2585:
2582:
2578:
2575:
2569:
2566:
2536:
2530:
2527:
2522:
2519:
2513:
2510:
2469:
2465:
2461:
2458:
2455:
2452:
2420:
2416:
2412:
2409:
2406:
2403:
2378:
2374:
2370:
2367:
2364:
2342:
2338:
2334:
2331:
2328:
2304:
2301:
2274:
2268:
2265:
2258:
2254:
2247:
2243:
2239:
2233:
2226:
2222:
2215:
2211:
2207:
2201:
2194:
2190:
2183:
2179:
2175:
2167:
2164:
2161:
2158:
2155:
2151:
2145:
2139:
2136:
2131:
2128:
2103:
2100:
2093:
2089:
2085:
2080:
2073:
2069:
2065:
2060:
2053:
2049:
2045:
2040:
2035:
2032:
2010:
2007:
2002:
1998:
1994:
1989:
1985:
1981:
1976:
1972:
1968:
1965:
1931:
1924:
1921:
1916:
1913:
1907:
1904:
1901:
1898:
1895:
1891:
1885:
1880:
1876:
1873:
1867:
1862:
1859:
1854:
1851:
1831:
1826:
1803:
1800:
1797:
1794:
1791:
1788:
1784:
1781:
1778:
1772:
1769:
1764:
1761:
1731:
1725:
1722:
1718:
1713:
1710:
1698:
1695:
1673:
1670:
1663:
1660:
1657:
1654:
1622:
1619:
1612:
1609:
1604:
1601:
1586:
1585:
1573:
1563:
1551:
1548:
1538:
1526:
1516:
1504:
1484:
1481:
1458:
1455:
1452:
1447:
1443:
1440:
1437:
1431:
1428:
1425:
1422:
1403:
1402:
1385:
1380:
1369:
1357:
1328:
1323:
1319:
1315:
1293:
1267:
1261:
1258:
1236:
1235:
1222:
1217:
1213:
1210:
1207:
1186:
1160:
1157:
1135:
1112:
1107:
1103:
1099:
1077:
1046:
1034:
1031:
1009:
1000:
996:
992:
987:
982:
978:
971:
963:
959:
955:
950:
945:
941:
934:
926:
922:
918:
913:
908:
904:
896:
892:
889:
883:
880:
875:
872:
844:
838:
835:
830:
827:
821:
818:
815:
810:
806:
776:
752:
727:
716:
715:
703:
700:
690:
670:
660:
643:
615:
612:
609:
606:
603:
600:
596:
575:
572:
569:
549:
525:
508:
505:
485:flow rates or
461:
458:
430:
427:
420:The theory of
417:
414:
385:Heisler Charts
331:Main article:
328:
325:
295:
292:
224:
220:
209:Knudsen number
173:electron fluid
161:metallic bonds
122:
121:
118:
115:
112:
109:
98:
97:
85:
82:
81:
80:
72:
66:
15:
9:
6:
4:
3:
2:
5483:
5482:
5471:
5468:
5466:
5463:
5461:
5460:Heat transfer
5458:
5456:
5453:
5452:
5450:
5440:
5436:
5432:
5429:
5426:
5423:
5420:
5415:
5411:
5410:
5402:
5399:
5397:
5393:
5389:
5386:
5383:
5380:
5377:
5374:
5373:
5366:
5365:0-88318-797-3
5362:
5358:
5352:
5337:
5331:
5327:
5326:
5318:
5303:
5297:
5293:
5292:
5284:
5269:
5263:
5259:
5258:
5250:
5235:
5229:
5225:
5224:
5216:
5208:
5202:
5198:
5197:
5189:
5182:
5176:
5168:
5164:
5160:
5158:9780470501979
5154:
5150:
5146:
5139:
5131:
5127:
5123:
5119:
5115:
5111:
5107:
5100:
5093:. 2016-08-22.
5092:
5088:
5082:
5068:
5064:
5058:
5056:
5051:
5041:
5038:
5036:
5033:
5031:
5028:
5026:
5023:
5021:
5018:
5016:
5013:
5011:
5008:
5006:
5003:
5001:
4998:
4996:
4993:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4971:
4967:
4963:
4955:
4952:
4947:
4933:
4931:
4927:
4926:heat transfer
4922:
4920:
4915:
4912:
4910:
4905:
4903:
4893:
4891:
4887:
4883:
4879:
4875:
4871:
4867:
4863:
4853:
4834:
4830:
4826:
4822:
4817:
4814:
4794:
4786:
4782:
4777:
4763:
4756:
4753:
4750:
4744:
4740:
4734:
4730:
4726:
4723:
4715:
4712:
4709:
4704:
4699:
4691:
4687:
4680:
4675:
4671:
4667:
4661:
4658:
4655:
4649:
4625:
4622:
4619:
4613:
4590:
4587:
4564:
4561:
4538:
4535:
4532:
4512:
4509:
4506:
4486:
4483:
4480:
4460:
4457:
4454:
4446:
4445:Splat cooling
4440:Splat cooling
4429:
4412:
4408:
4399:
4396:
4393:
4389:
4385:
4382:
4381:
4380:
4377:
4364:
4357:
4353:
4348:
4345:
4339:
4323:
4318:
4313:
4291:
4271:
4266:
4261:
4248:
4244:
4238:
4233:
4229:
4225:
4220:
4217:
4214:
4211:
4205:
4201:
4198:
4195:
4187:
4183:
4179:
4174:
4170:
4162:
4158:
4154:
4151:
4128:
4125:
4118:
4115:
4112:
4109:
4078:
4047:
4025:
4021:
4018:
4012:
3990:
3987:
3979:
3969:
3965:
3961:
3955:
3954:
3950:
3945:This section
3943:
3939:
3934:
3933:
3924:
3923:Heat equation
3914:
3896:
3892:
3888:
3883:
3879:
3871:
3867:
3861:
3857:
3848:
3844:
3840:
3835:
3831:
3821:
3818:
3815:
3812:
3803:
3799:
3793:
3789:
3786:
3780:
3776:
3770:
3766:
3759:
3755:
3751:
3746:
3742:
3735:
3732:
3729:
3726:
3720:
3717:
3705:
3692:
3687:
3683:
3679:
3676:
3673:
3670:
3662:
3657:
3641:
3637:
3614:
3610:
3595:
3573:
3569:
3564:
3558:
3554:
3547:
3544:
3537:
3533:
3529:
3524:
3520:
3513:
3508:
3504:
3478:
3474:
3470:
3465:
3461:
3453:
3449:
3445:
3440:
3436:
3427:
3423:
3419:
3416:
3413:
3410:
3407:
3401:
3398:
3372:
3369:
3366:
3363:
3353:
3349:
3344:
3338:
3334:
3327:
3324:
3318:
3311:
3308:
3302:
3293:
3288:
3284:
3255:
3251:
3246:
3240:
3236:
3229:
3226:
3219:
3215:
3211:
3206:
3202:
3195:
3192:
3189:
3186:
3183:
3177:
3174:
3151:
3148:
3141:
3137:
3129:
3125:
3120:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3089:
3086:
3077:
3073:
3065:
3061:
3056:
3049:
3046:
3020:
3017:
3012:
3009:
3003:
3000:
2997:
2994:
2991:
2988:
2985:
2979:
2976:
2971:
2968:
2960:
2956:
2952:
2949:
2946:
2940:
2937:
2925:
2912:
2909:
2906:
2903:
2900:
2895:
2891:
2881:
2865:
2861:
2857:
2852:
2848:
2827:
2805:
2801:
2778:
2774:
2759:
2756:
2743:
2740:
2736:
2732:
2729:
2726:
2706:
2699:
2696:
2690:
2681:
2678:
2670:
2665:
2652:
2649:
2646:
2643:
2640:
2620:
2616:
2612:
2609:
2606:
2586:
2583:
2576:
2573:
2567:
2564:
2552:
2534:
2528:
2520:
2517:
2511:
2508:
2499:
2497:
2493:
2489:
2485:
2467:
2463:
2459:
2456:
2453:
2450:
2442:
2438:
2434:
2418:
2414:
2410:
2407:
2404:
2401:
2392:
2376:
2372:
2368:
2365:
2362:
2340:
2336:
2332:
2329:
2326:
2318:
2314:
2310:
2300:
2298:
2294:
2290:
2285:
2272:
2266:
2263:
2256:
2252:
2245:
2241:
2231:
2224:
2220:
2213:
2209:
2199:
2192:
2188:
2181:
2177:
2162:
2156:
2149:
2143:
2137:
2129:
2114:
2101:
2098:
2091:
2087:
2083:
2078:
2071:
2067:
2063:
2058:
2051:
2047:
2043:
2038:
2033:
2030:
2008:
2005:
2000:
1996:
1992:
1987:
1983:
1979:
1974:
1970:
1966:
1963:
1954:
1948:
1942:
1929:
1922:
1914:
1902:
1896:
1889:
1883:
1878:
1874:
1865:
1860:
1857:
1852:
1849:
1842:is given by:
1829:
1814:
1801:
1795:
1789:
1782:
1779:
1776:
1770:
1762:
1747:
1729:
1723:
1716:
1711:
1708:
1694:
1692:
1691:heat equation
1687:
1671:
1668:
1661:
1658:
1655:
1644:
1641:for a simple
1640:
1635:
1620:
1617:
1610:
1607:
1602:
1599:
1591:
1571:
1564:
1549:
1539:
1524:
1517:
1502:
1482:
1472:
1471:
1470:
1456:
1453:
1445:
1441:
1435:
1429:
1426:
1423:
1420:
1412:
1408:
1400:
1370:
1355:
1348:
1344:
1343:thermal power
1317:
1291:
1265:
1259:
1256:
1245:
1244:
1243:
1241:
1211:
1208:
1184:
1158:
1155:
1133:
1101:
1075:
1060:
1059:
1058:
1044:
1033:Integral form
1030:
1028:
1024:
1007:
998:
994:
985:
980:
969:
961:
957:
948:
943:
932:
924:
920:
911:
906:
894:
890:
887:
881:
873:
860:
859:heat equation
855:
842:
836:
833:
828:
825:
819:
816:
813:
808:
804:
794:
788:
774:
766:
750:
742:
725:
701:
691:
688:
684:
668:
661:
658:
633:
632:
631:
629:
613:
610:
604:
601:
598:
573:
567:
547:
540:
514:
504:
502:
498:
494:
490:
488:
483:
479:
475:
474:heat transfer
471:
470:heat equation
467:
466:Fourier's law
460:Fourier's law
457:
455:
451:
447:
443:
442:heat transfer
439:
435:
426:
423:
413:
410:
405:
401:
400:heat capacity
397:
393:
388:
386:
381:
377:
376:heat equation
371:
367:
363:
361:
355:
351:
347:
344:
340:
334:
333:Heat equation
324:
322:
318:
313:
309:
306:
302:
291:
289:
285:
281:
280:
275:
271:
267:
263:
259:
255:
251:
247:
243:
238:
222:
218:
210:
204:
202:
198:
194:
190:
186:
182:
178:
174:
170:
166:
162:
158:
153:
151:
147:
143:
138:
135:
130:
126:
119:
116:
113:
110:
107:
106:
105:
96:
91:
90:Heat equation
77:
73:
70:
67:
63:
60:
59:
58:
55:
53:
49:
45:
41:
33:
28:
22:
5356:
5351:
5339:. Retrieved
5324:
5317:
5305:. Retrieved
5290:
5283:
5271:. Retrieved
5256:
5249:
5237:. Retrieved
5222:
5215:
5195:
5188:
5175:
5148:
5138:
5113:
5109:
5099:
5090:
5081:
5070:. Retrieved
5066:
4968:
4964:
4961:
4948:
4944:
4923:
4916:
4913:
4906:
4899:
4882:Biot numbers
4859:
4778:
4443:
4435:Applications
4391:
4387:
4378:
4321:
4314:
4269:
4262:
3997:
3982:
3973:
3958:Please help
3946:
3706:
3661:surface area
3658:
3601:
2926:
2882:
2765:
2757:
2671:, given by:
2668:
2666:
2553:
2500:
2495:
2491:
2487:
2483:
2440:
2436:
2432:
2393:
2306:
2286:
2115:
1952:
1946:
1943:
1815:
1748:
1700:
1688:
1636:
1587:
1404:
1237:
1036:
856:
792:
789:
717:
683:conductivity
510:
491:
478:proportional
463:
434:Second sound
432:
419:
403:
389:
372:
368:
364:
356:
352:
348:
342:
338:
336:
314:
310:
304:
300:
297:
277:
273:
269:
265:
261:
257:
253:
245:
239:
205:
172:
154:
152:vibrations.
139:
127:
123:
103:
93:
75:
68:
61:
56:
31:
26:
25:
5116:: 500â505.
5025:Biot number
4936:Instruments
1697:Conductance
1027:heat kernel
796:direction:
741:anisotropic
169:ionic bonds
48:temperature
5449:Categories
5367:, page 23.
5072:2024-08-19
5046:References
4958:Gas sensor
4930:symmetries
4874:martensite
2501:For heat,
2293:turbulence
1411:integrated
1405:The above
444:occurs by
177:conductive
157:conductors
146:insulators
88:See also:
69:Convection
62:Conduction
5167:713621645
5130:114611104
5000:Heat pipe
4870:austenite
4862:quenching
4827:ρ
4815:α
4795:α
4754:α
4735:−
4727:
4713:α
4710:π
4697:Δ
4668:−
4629:∞
4626:≤
4620:≤
4617:∞
4614:−
4594:∞
4568:∞
4542:∞
4539:−
4346:α
4226:ρ
4212:−
4202:
4180:−
4155:−
4123:Δ
4116:−
3947:does not
3889:−
3841:−
3822:π
3787:−
3752:−
3736:π
3721:˙
3680:π
3598:Spherical
3548:
3530:−
3471:−
3446:−
3420:ℓ
3414:π
3402:˙
3373:ℓ
3367:π
3328:
3312:˙
3300:Δ
3230:
3212:−
3196:ℓ
3193:π
3178:˙
3121:∫
3117:ℓ
3114:π
3105:−
3057:∫
3050:˙
3004:ℓ
2998:π
2989:−
2950:−
2941:˙
2913:ℓ
2907:π
2858:−
2828:ℓ
2700:˙
2688:Δ
2581:Δ
2568:˙
2526:Δ
2408:ρ
2317:Ohm's law
2297:viscosity
2289:thin film
2267:⋯
2238:Δ
2206:Δ
2174:Δ
2160:Δ
2157:−
2135:Δ
2127:Δ
2102:⋯
2009:⋯
1920:Δ
1912:Δ
1900:Δ
1897:−
1872:Δ
1793:Δ
1790:−
1768:Δ
1760:Δ
1721:Δ
1672:˙
1653:Δ
1639:Ohm's law
1547:Δ
1480:Δ
1451:Δ
1439:Δ
1427:−
1318:⋅
1260:˙
1212:⋅
1206:∇
1156:−
1102:⋅
991:∂
977:∂
954:∂
940:∂
917:∂
903:∂
891:α
879:∂
871:∂
817:−
699:∇
608:∇
602:−
571:∇
568:−
513:heat flux
497:Ohm's law
450:diffusion
404:variation
201:diffusion
76:radiation
4974:See also
4890:nomogram
4878:eutectic
4400:Convert
4317:nomogram
3976:May 2013
3496:, where
2482:, where
2431:, where
1701:Writing
1242:units):
515:density
482:gradient
279:property
165:covalent
84:Overview
44:hotplate
3968:removed
3953:sources
1409:, when
1401:(in m).
1368:(in J),
1341:is the
1021:with a
685:, W/(m·
630:units)
339:in time
185:thermal
104:Where:
5394:
5363:
5332:
5298:
5264:
5230:
5203:
5165:
5155:
5128:
4860:Metal
4783:. The
2547:where
1742:where
1469:where
1397:is an
765:tensor
487:fluxes
193:copper
150:phonon
101:ÎșAÎT/â
5341:7 May
5307:7 May
5273:7 May
5239:7 May
5126:S2CID
436:is a
284:phase
248:. In
175:of a
79:body.
5392:ISBN
5361:ISBN
5343:2013
5330:ISBN
5309:2013
5296:ISBN
5275:2013
5262:ISBN
5241:2013
5228:ISBN
5201:ISBN
5179:The
5163:OCLC
5153:ISBN
4606:for
4554:and
4499:and
4263:The
3951:any
3949:cite
3659:The
3388:and
2295:and
1950:and
1347:heat
446:wave
187:and
5118:doi
5114:846
4872:to
4724:exp
4525:at
4390:or
4199:exp
3962:by
2633:or
659:/m,
167:or
5451::
5437:,
5161:.
5124:.
5112:.
5108:.
5089:.
5065:.
5054:^
4907:A
4335:Fo
4267:,
4008:Bi
3545:ln
3325:ln
3227:ln
2880:.
2391:.
1693:.
1645::
1240:SI
1029:.
689:),
628:SI
387:.
252:,
237:.
5441:.
5345:.
5311:.
5277:.
5243:.
5209:.
5169:.
5132:.
5120::
5075:.
4835:p
4831:C
4823:k
4818:=
4764:)
4757:t
4751:4
4745:2
4741:x
4731:(
4716:t
4705:2
4700:X
4692:i
4688:T
4681:=
4676:i
4672:T
4665:)
4662:t
4659:,
4656:x
4653:(
4650:T
4623:x
4591:=
4588:t
4565:=
4562:x
4536:=
4533:x
4513:0
4510:=
4507:T
4487:0
4484:=
4481:x
4461:0
4458:=
4455:t
4413:i
4409:T
4394:.
4392:L
4388:x
4365:.
4358:2
4354:L
4349:t
4340:=
4322:y
4297:K
4292:2
4288:m
4283:W
4270:h
4249:.
4245:)
4239:V
4234:p
4230:C
4221:t
4218:A
4215:h
4206:(
4196:=
4188:f
4184:T
4175:i
4171:T
4163:f
4159:T
4152:T
4129:,
4126:T
4119:h
4113:=
4110:q
4087:K
4084:s
4079:2
4075:m
4070:J
4048:h
4026:k
4022:L
4019:h
4013:=
3989:)
3983:(
3978:)
3974:(
3970:.
3956:.
3897:1
3893:r
3884:2
3880:r
3872:2
3868:r
3862:1
3858:r
3854:)
3849:2
3845:T
3836:1
3832:T
3828:(
3819:k
3816:4
3813:=
3804:2
3800:r
3794:/
3790:1
3781:1
3777:r
3771:/
3767:1
3760:2
3756:T
3747:1
3743:T
3733:k
3730:4
3727:=
3718:Q
3693:.
3688:2
3684:r
3677:4
3674:=
3671:A
3642:2
3638:r
3615:1
3611:r
3579:)
3574:1
3570:r
3565:/
3559:2
3555:r
3551:(
3538:1
3534:r
3525:2
3521:r
3514:=
3509:m
3505:r
3479:1
3475:r
3466:2
3462:r
3454:2
3450:T
3441:1
3437:T
3428:m
3424:r
3417:k
3411:2
3408:=
3399:Q
3370:k
3364:2
3359:)
3354:1
3350:r
3345:/
3339:2
3335:r
3331:(
3319:=
3309:Q
3303:T
3294:=
3289:c
3285:R
3261:)
3256:1
3252:r
3247:/
3241:2
3237:r
3233:(
3220:2
3216:T
3207:1
3203:T
3190:k
3187:2
3184:=
3175:Q
3152:T
3149:d
3142:2
3138:T
3130:1
3126:T
3111:k
3108:2
3102:=
3099:r
3096:d
3090:r
3087:1
3078:2
3074:r
3066:1
3062:r
3047:Q
3021:r
3018:d
3013:T
3010:d
3001:r
2995:k
2992:2
2986:=
2980:r
2977:d
2972:T
2969:d
2961:r
2957:A
2953:k
2947:=
2938:Q
2910:r
2904:2
2901:=
2896:r
2892:A
2866:1
2862:T
2853:2
2849:T
2806:2
2802:r
2779:1
2775:r
2744:.
2741:I
2737:/
2733:V
2730:=
2727:R
2707:,
2697:Q
2691:T
2682:=
2679:R
2669:R
2653:.
2650:G
2647:V
2644:=
2641:I
2621:R
2617:/
2613:V
2610:=
2607:I
2587:,
2584:T
2577:U
2574:=
2565:Q
2549:U
2535:,
2529:x
2521:A
2518:k
2512:=
2509:U
2496:A
2492:x
2488:k
2484:G
2468:x
2464:/
2460:A
2457:k
2454:=
2451:G
2441:A
2437:x
2433:Ï
2419:A
2415:/
2411:x
2405:=
2402:R
2377:V
2373:/
2369:I
2366:=
2363:G
2341:I
2337:/
2333:V
2330:=
2327:R
2273:.
2264:+
2257:3
2253:k
2246:3
2242:x
2232:+
2225:2
2221:k
2214:2
2210:x
2200:+
2193:1
2189:k
2182:1
2178:x
2166:)
2163:T
2154:(
2150:A
2144:=
2138:t
2130:Q
2099:+
2092:3
2088:U
2084:1
2079:+
2072:2
2068:U
2064:1
2059:+
2052:1
2048:U
2044:1
2039:=
2034:U
2031:1
2006:+
2001:3
1997:R
1993:+
1988:2
1984:R
1980:+
1975:1
1971:R
1967:=
1964:R
1953:Q
1947:A
1930:.
1923:t
1915:Q
1906:)
1903:T
1894:(
1890:A
1884:=
1879:k
1875:x
1866:=
1861:U
1858:1
1853:=
1850:R
1830:R
1802:.
1799:)
1796:T
1787:(
1783:A
1780:U
1777:=
1771:t
1763:Q
1744:U
1730:,
1724:x
1717:k
1712:=
1709:U
1669:Q
1662:R
1659:=
1656:T
1621:A
1618:L
1611:k
1608:1
1603:=
1600:R
1572:L
1550:T
1525:A
1503:Q
1483:t
1457:,
1454:T
1446:L
1442:t
1436:A
1430:k
1424:=
1421:Q
1384:S
1379:d
1356:Q
1327:S
1322:d
1314:q
1292:S
1266:=
1257:Q
1221:S
1216:d
1209:T
1185:S
1159:k
1134:=
1111:S
1106:d
1098:q
1076:S
1045:S
1008:)
999:2
995:z
986:T
981:2
970:+
962:2
958:y
949:T
944:2
933:+
925:2
921:x
912:T
907:2
895:(
888:=
882:t
874:T
843:.
837:x
834:d
829:T
826:d
820:k
814:=
809:x
805:q
793:x
775:k
751:k
726:k
702:T
687:K
669:k
657:W
642:q
614:,
611:T
605:k
599:=
595:q
574:T
548:k
524:q
274:T
270:A
266:L
262:t
258:Q
254:k
246:k
223:n
219:K
36:k
23:.
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